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Theorem fin23lem14 9747
Description: Lemma for fin23 9803. 𝑈 will never evolve to an empty set if it did not start with one. (Contributed by Stefan O'Rear, 1-Nov-2014.)
Hypothesis
Ref Expression
fin23lem.a 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
Assertion
Ref Expression
fin23lem14 ((𝐴 ∈ ω ∧ ran 𝑡 ≠ ∅) → (𝑈𝐴) ≠ ∅)
Distinct variable groups:   𝑡,𝑖,𝑢   𝐴,𝑖,𝑢   𝑈,𝑖,𝑢
Allowed substitution hints:   𝐴(𝑡)   𝑈(𝑡)

Proof of Theorem fin23lem14
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6663 . . . . 5 (𝑎 = ∅ → (𝑈𝑎) = (𝑈‘∅))
21neeq1d 3073 . . . 4 (𝑎 = ∅ → ((𝑈𝑎) ≠ ∅ ↔ (𝑈‘∅) ≠ ∅))
32imbi2d 343 . . 3 (𝑎 = ∅ → (( ran 𝑡 ≠ ∅ → (𝑈𝑎) ≠ ∅) ↔ ( ran 𝑡 ≠ ∅ → (𝑈‘∅) ≠ ∅)))
4 fveq2 6663 . . . . 5 (𝑎 = 𝑏 → (𝑈𝑎) = (𝑈𝑏))
54neeq1d 3073 . . . 4 (𝑎 = 𝑏 → ((𝑈𝑎) ≠ ∅ ↔ (𝑈𝑏) ≠ ∅))
65imbi2d 343 . . 3 (𝑎 = 𝑏 → (( ran 𝑡 ≠ ∅ → (𝑈𝑎) ≠ ∅) ↔ ( ran 𝑡 ≠ ∅ → (𝑈𝑏) ≠ ∅)))
7 fveq2 6663 . . . . 5 (𝑎 = suc 𝑏 → (𝑈𝑎) = (𝑈‘suc 𝑏))
87neeq1d 3073 . . . 4 (𝑎 = suc 𝑏 → ((𝑈𝑎) ≠ ∅ ↔ (𝑈‘suc 𝑏) ≠ ∅))
98imbi2d 343 . . 3 (𝑎 = suc 𝑏 → (( ran 𝑡 ≠ ∅ → (𝑈𝑎) ≠ ∅) ↔ ( ran 𝑡 ≠ ∅ → (𝑈‘suc 𝑏) ≠ ∅)))
10 fveq2 6663 . . . . 5 (𝑎 = 𝐴 → (𝑈𝑎) = (𝑈𝐴))
1110neeq1d 3073 . . . 4 (𝑎 = 𝐴 → ((𝑈𝑎) ≠ ∅ ↔ (𝑈𝐴) ≠ ∅))
1211imbi2d 343 . . 3 (𝑎 = 𝐴 → (( ran 𝑡 ≠ ∅ → (𝑈𝑎) ≠ ∅) ↔ ( ran 𝑡 ≠ ∅ → (𝑈𝐴) ≠ ∅)))
13 vex 3496 . . . . . . 7 𝑡 ∈ V
1413rnex 7609 . . . . . 6 ran 𝑡 ∈ V
1514uniex 7459 . . . . 5 ran 𝑡 ∈ V
16 fin23lem.a . . . . . 6 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
1716seqom0g 8084 . . . . 5 ( ran 𝑡 ∈ V → (𝑈‘∅) = ran 𝑡)
1815, 17mp1i 13 . . . 4 ( ran 𝑡 ≠ ∅ → (𝑈‘∅) = ran 𝑡)
19 id 22 . . . 4 ( ran 𝑡 ≠ ∅ → ran 𝑡 ≠ ∅)
2018, 19eqnetrd 3081 . . 3 ( ran 𝑡 ≠ ∅ → (𝑈‘∅) ≠ ∅)
2116fin23lem12 9745 . . . . . . 7 (𝑏 ∈ ω → (𝑈‘suc 𝑏) = if(((𝑡𝑏) ∩ (𝑈𝑏)) = ∅, (𝑈𝑏), ((𝑡𝑏) ∩ (𝑈𝑏))))
2221adantr 483 . . . . . 6 ((𝑏 ∈ ω ∧ (𝑈𝑏) ≠ ∅) → (𝑈‘suc 𝑏) = if(((𝑡𝑏) ∩ (𝑈𝑏)) = ∅, (𝑈𝑏), ((𝑡𝑏) ∩ (𝑈𝑏))))
23 iftrue 4471 . . . . . . . . 9 (((𝑡𝑏) ∩ (𝑈𝑏)) = ∅ → if(((𝑡𝑏) ∩ (𝑈𝑏)) = ∅, (𝑈𝑏), ((𝑡𝑏) ∩ (𝑈𝑏))) = (𝑈𝑏))
2423adantr 483 . . . . . . . 8 ((((𝑡𝑏) ∩ (𝑈𝑏)) = ∅ ∧ (𝑏 ∈ ω ∧ (𝑈𝑏) ≠ ∅)) → if(((𝑡𝑏) ∩ (𝑈𝑏)) = ∅, (𝑈𝑏), ((𝑡𝑏) ∩ (𝑈𝑏))) = (𝑈𝑏))
25 simprr 771 . . . . . . . 8 ((((𝑡𝑏) ∩ (𝑈𝑏)) = ∅ ∧ (𝑏 ∈ ω ∧ (𝑈𝑏) ≠ ∅)) → (𝑈𝑏) ≠ ∅)
2624, 25eqnetrd 3081 . . . . . . 7 ((((𝑡𝑏) ∩ (𝑈𝑏)) = ∅ ∧ (𝑏 ∈ ω ∧ (𝑈𝑏) ≠ ∅)) → if(((𝑡𝑏) ∩ (𝑈𝑏)) = ∅, (𝑈𝑏), ((𝑡𝑏) ∩ (𝑈𝑏))) ≠ ∅)
27 iffalse 4474 . . . . . . . . 9 (¬ ((𝑡𝑏) ∩ (𝑈𝑏)) = ∅ → if(((𝑡𝑏) ∩ (𝑈𝑏)) = ∅, (𝑈𝑏), ((𝑡𝑏) ∩ (𝑈𝑏))) = ((𝑡𝑏) ∩ (𝑈𝑏)))
2827adantr 483 . . . . . . . 8 ((¬ ((𝑡𝑏) ∩ (𝑈𝑏)) = ∅ ∧ (𝑏 ∈ ω ∧ (𝑈𝑏) ≠ ∅)) → if(((𝑡𝑏) ∩ (𝑈𝑏)) = ∅, (𝑈𝑏), ((𝑡𝑏) ∩ (𝑈𝑏))) = ((𝑡𝑏) ∩ (𝑈𝑏)))
29 neqne 3022 . . . . . . . . 9 (¬ ((𝑡𝑏) ∩ (𝑈𝑏)) = ∅ → ((𝑡𝑏) ∩ (𝑈𝑏)) ≠ ∅)
3029adantr 483 . . . . . . . 8 ((¬ ((𝑡𝑏) ∩ (𝑈𝑏)) = ∅ ∧ (𝑏 ∈ ω ∧ (𝑈𝑏) ≠ ∅)) → ((𝑡𝑏) ∩ (𝑈𝑏)) ≠ ∅)
3128, 30eqnetrd 3081 . . . . . . 7 ((¬ ((𝑡𝑏) ∩ (𝑈𝑏)) = ∅ ∧ (𝑏 ∈ ω ∧ (𝑈𝑏) ≠ ∅)) → if(((𝑡𝑏) ∩ (𝑈𝑏)) = ∅, (𝑈𝑏), ((𝑡𝑏) ∩ (𝑈𝑏))) ≠ ∅)
3226, 31pm2.61ian 810 . . . . . 6 ((𝑏 ∈ ω ∧ (𝑈𝑏) ≠ ∅) → if(((𝑡𝑏) ∩ (𝑈𝑏)) = ∅, (𝑈𝑏), ((𝑡𝑏) ∩ (𝑈𝑏))) ≠ ∅)
3322, 32eqnetrd 3081 . . . . 5 ((𝑏 ∈ ω ∧ (𝑈𝑏) ≠ ∅) → (𝑈‘suc 𝑏) ≠ ∅)
3433ex 415 . . . 4 (𝑏 ∈ ω → ((𝑈𝑏) ≠ ∅ → (𝑈‘suc 𝑏) ≠ ∅))
3534imim2d 57 . . 3 (𝑏 ∈ ω → (( ran 𝑡 ≠ ∅ → (𝑈𝑏) ≠ ∅) → ( ran 𝑡 ≠ ∅ → (𝑈‘suc 𝑏) ≠ ∅)))
363, 6, 9, 12, 20, 35finds 7600 . 2 (𝐴 ∈ ω → ( ran 𝑡 ≠ ∅ → (𝑈𝐴) ≠ ∅))
3736imp 409 1 ((𝐴 ∈ ω ∧ ran 𝑡 ≠ ∅) → (𝑈𝐴) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398   = wceq 1530  wcel 2107  wne 3014  Vcvv 3493  cin 3933  c0 4289  ifcif 4465   cuni 4830  ran crn 5549  suc csuc 6186  cfv 6348  cmpo 7150  ωcom 7572  seqωcseqom 8075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-seqom 8076
This theorem is referenced by:  fin23lem21  9753
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